Homotopy Classification of Gerbes

Abstract

Gerbes are locally connected presheaves of groupoids on a small Grothendieck site $\mathcal{C}$. They are classified up to local weak equivalence by path components of a cocycle category taking values in the big $2$-groupoid $\mathbf{Iso}(\Gr(\mathcal{C}))$ consisting of all sheaves of groups on $\mathcal{C}$, their isomorphisms and homotopies. If $\mathcal{F}$ is a full subpresheaf of $\mathbf{Iso}(\Gr(\mathcal{C}))$ then the set $[\ast,B\mathcal{F}]$ of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally weakly equivalent to objects of $\mathcal{F}$. If $\St(\pi \mathcal{F})$ is the stack completion of the fundamental groupoid $\pi\mathcal{F}$ of $\mathcal{F}$, if $L$ is a global section of $\St(\pi\mathcal{F})$, and if $F_{L}$ is the homotopy fibre over $L$ of the canonical map $B\mathcal{F} \to B\St(\pi\mathcal{F})$, then $[\ast,F_{L}]$ is in bijective correspondence with Giraud's non-abelian cohomology object $H^{2}(\mathcal{C},L)$ of equivalence classes of gerbes with band $L$.